

A189759


Numbers pqr such that pq + pr + qr is prime, where p, q, and r are primes.


3



30, 42, 66, 70, 78, 105, 114, 130, 154, 165, 174, 182, 222, 231, 238, 246, 255, 273, 282, 285, 286, 310, 318, 345, 357, 366, 370, 385, 399, 418, 430, 434, 442, 455, 465, 474, 483, 494, 498, 518, 555, 561, 574, 582, 595, 602, 609, 618, 642, 645, 651, 663, 665
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OFFSET

1,1


COMMENTS

The number pq+pr+qr is prime only if p, q, and r are distinct. The primes of form pq+pr+qr are in A087054. A prime may have multiple representations as pq+pr+qr; for example, 2*3*13 and 3*5*7 both produce the prime 71.
As mentioned by Ufnarovski and Ahlander, if pq+pr+qr is prime, then the arithmetic derivative (A003415) of pqr is that prime. They conjecture that this sequence and A087054 are infinite.


LINKS



MATHEMATICA

pqr[nn_] := Module[{p=Prime[Range[PrimePi[nn/6]+1]], i, j, k, n, prod}, Sort[Reap[i=0; While[i++; p[[i]]p[[i+1]]p[[i+2]] <= nn, j=i; While[j++; p[[i]]p[[j]]p[[j+1]] <= nn, k=j; While[k++; prod=p[[i]]p[[j]]p[[k]]; prod <= nn, n=p[[i]]p[[j]]+p[[i]]p[[k]]+p[[j]]p[[k]]; If[PrimeQ[n], Sow[prod]]]]]][[2, 1]]]]; pqr[1000]
Take[Union[Times@@@Select[Subsets[Prime[Range[30]], {3}], PrimeQ[ Total[ Times@@@Subsets[#, {2}]]]&]], 60](* Harvey P. Dale, Dec 29 2011 *)


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



